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Strict saddle critical point

PDF p. 3: Section 1, strict-saddle parenthesis; PDF p. 4: Section 1, strict-saddle parenthesis

Definition

Let $G = (V, E)$ and its homogeneous Kuramoto energy $E_{G}$ be as in Homogeneous Kuramoto energy, on the phase torus $T_{V} = (S^{1})^{V}$ . In angular coordinates, a critical point is a state $θ$ at which every first partial derivative of $E_{G}$ is zero. Its Hessian $\nabla^{2} E_{G} (θ)$ is the real matrix of second partial derivatives in those coordinates. An eigenvalue $λ$ of this matrix is a scalar for which $\nabla^{2} E_{G} (θ) v = λ v$ for some nonzero vector $v$ .

A strict saddle is a critical point $θ$ for which $\nabla^{2} E_{G} (θ)$ has a strictly negative eigenvalue.

Facts & Assumptions

[F1]

The energy is the finite sum $E_{G} (θ) = \frac{1}{2} \sum_{x, y \in V} A_{x, y} (1 - \cos (θ_{x} - θ_{y}))$ , evaluated using real lifts (Homogeneous Kuramoto energy).

[F2]

Angular chart transitions on the phase torus are locally translations by vectors in $(2 π Z)^{V}$ (Finite phase tori, local lifts and the flat metric).

[F3]

Sine and cosine and their stated derivatives are periodic under shifts by integer multiples of $2 π$ ; the derivative and finite-sum rules apply (Phasors, trigonometric identities and periodic derivatives).

[F4]

The gradient and Hessian consist respectively of first and second partial derivatives, and a Euclidean critical point has zero gradient (The Hessian matrix and critical points of a scalar field).

[F5]

For real functions $f, g$ , if $f$ is differentiable at $c$ , $g$ is differentiable at $f (c)$ , and $g \circ f$ is defined near $c$ , then $(g \circ f)^{'} (c) = g^{'} (f (c)) f^{'} (c)$ . The sum and product rules also hold (Sum, product, quotient, and chain rules from difference quotients).

Justification

Given: A phase state $θ$ and angular charts containing it.

Goal: Show that the critical-point and negative-eigenvalue conditions do not depend on the chart or lift.

1.1 Holding all but one angular coordinate fixed makes each phase difference in [F1] an affine real function. The chain, sum and product rules in [F5], together with the trigonometric derivatives in [F3], therefore give the first partial derivatives of the finite energy expression. Differentiating once more in any angular coordinate uses the same rules and gives all second partial derivatives. By [F2], two chart coordinates near $θ$ differ by a constant lattice translation. The periodicity in [F3] and these differentiation rules show that the corresponding first and second partial derivatives agree at $θ$ . [F1, F2, F3, F5]

2.1 By [F4] and step 1.1, the gradient and Hessian have the same entries in either chart. Thus zero gradient is independent of the chart, and the Hessian has the same eigenvalues in either chart. The stated conditions therefore define an unambiguous subset of the phase torus. [F4, step 1.1] ∎

Source notes

The strict-saddle terminology is attributed to Expander graphs are globally synchronizing, PDF pp. 3–4, Section 1. The justification uses the mathematical suppliers above.

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Sources

Expander graphs are globally synchronizing · PDF pp. 3–4, Section 1, strict-saddle parenthesis